Certain new bounds considering the weighted Simpson-like type inequality and applications

Area of Science:

• Mathematical Analysis
• Inequalities
• Probability Theory

Background:

• The study builds upon existing Simpson-type inequalities, which are fundamental in numerical integration and analysis.
• Weighted inequalities are crucial for handling non-uniform data distributions and improving approximation accuracy.
• First-order differentiable mappings are a key class of functions in various mathematical and scientific domains.

Purpose of the Study:

• To establish a novel weighted Simpson-type identity.
• To derive new estimation-type results concerning weighted Simpson-like inequalities for differentiable functions.
• To explore the utility of these findings in applied areas such as information theory and statistics.

Main Methods:

• The research employs techniques from classical analysis and inequality theory.
• Specific focus is placed on the properties of first-order differentiable mappings.
• The derivation involves careful manipulation of integral forms and estimation bounds.

Main Results:

• A new weighted Simpson-type identity has been successfully formulated.
• Novel estimation bounds related to the weighted Simpson-like inequality were obtained.
• The results provide tighter estimations compared to existing inequalities under certain conditions.

Conclusions:

• The established weighted Simpson-type identity and associated inequalities offer valuable tools for mathematical analysis.
• The findings have direct implications for approximating f-divergence measures and calculating higher moments of random variables.
• This work contributes to the advancement of inequality theory with practical applications.